L(s) = 1 | − 0.942·2-s − 3-s − 1.11·4-s + 5-s + 0.942·6-s + 5.06·7-s + 2.93·8-s + 9-s − 0.942·10-s + 0.595·11-s + 1.11·12-s + 2.07·13-s − 4.77·14-s − 15-s − 0.539·16-s + 1.13·17-s − 0.942·18-s − 3.01·19-s − 1.11·20-s − 5.06·21-s − 0.561·22-s + 5.50·23-s − 2.93·24-s + 25-s − 1.95·26-s − 27-s − 5.63·28-s + ⋯ |
L(s) = 1 | − 0.666·2-s − 0.577·3-s − 0.555·4-s + 0.447·5-s + 0.384·6-s + 1.91·7-s + 1.03·8-s + 0.333·9-s − 0.298·10-s + 0.179·11-s + 0.320·12-s + 0.575·13-s − 1.27·14-s − 0.258·15-s − 0.134·16-s + 0.275·17-s − 0.222·18-s − 0.691·19-s − 0.248·20-s − 1.10·21-s − 0.119·22-s + 1.14·23-s − 0.598·24-s + 0.200·25-s − 0.383·26-s − 0.192·27-s − 1.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.646725796\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.646725796\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 + 0.942T + 2T^{2} \) |
| 7 | \( 1 - 5.06T + 7T^{2} \) |
| 11 | \( 1 - 0.595T + 11T^{2} \) |
| 13 | \( 1 - 2.07T + 13T^{2} \) |
| 17 | \( 1 - 1.13T + 17T^{2} \) |
| 19 | \( 1 + 3.01T + 19T^{2} \) |
| 23 | \( 1 - 5.50T + 23T^{2} \) |
| 29 | \( 1 - 3.69T + 29T^{2} \) |
| 31 | \( 1 - 5.05T + 31T^{2} \) |
| 37 | \( 1 - 3.37T + 37T^{2} \) |
| 41 | \( 1 - 0.973T + 41T^{2} \) |
| 43 | \( 1 - 0.997T + 43T^{2} \) |
| 47 | \( 1 - 5.67T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 0.309T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 7.89T + 79T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 - 4.28T + 89T^{2} \) |
| 97 | \( 1 + 6.88T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.136062586797177436198234304484, −7.61601775054412024690958215299, −6.74649877633504394037933476745, −5.84697264139364487085853748122, −5.09681539363009700172270453337, −4.62541526411970644846258446703, −3.95631623610926285405769212103, −2.45819858896399891175348400365, −1.40682585795618858325497019685, −0.919710250708581610456293076516,
0.919710250708581610456293076516, 1.40682585795618858325497019685, 2.45819858896399891175348400365, 3.95631623610926285405769212103, 4.62541526411970644846258446703, 5.09681539363009700172270453337, 5.84697264139364487085853748122, 6.74649877633504394037933476745, 7.61601775054412024690958215299, 8.136062586797177436198234304484